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Jul
1
comment Examples of calculus on “strange” spaces
Maybe you're looking for the term "derivation" in the sense of abstract algebra? Maybe you'd be interested in en.wikipedia.org/wiki/Arithmetic_derivative ?
Jul
1
comment Unusual result to the addition
@rogerl It's worth noting that that expression for a sequence of 1s can be found as an application of the formula for the sum of a finite geometric series.
Jul
1
comment Rational Irrational Numbers
As an aside, you may be interested in reading about continued fraction representations of numbers, where irrational solutions to quadratic equations with rational coefficients (like $\sqrt2$) have repeating expansions, but other irrationals don't.
Jul
1
comment “On Numbers and Games” or “Winning Ways for Your Mathematical Plays”?
I read those two simultaneously, but I would recommend Lessons in Play as a starter book over either of those. It's more accessible to an undergrad and generally clearer about theorems and proofs, for the most part. After that, if you want to read heavier/broader theory, Combinatorial Game Theory by Siegel would be a good option (covering essentially all of the theorems from ONAG and WW), but it doesn't have too many examples, being a graduate textbook.
Jun
19
comment Is there such a thing as “quadratic independence” (and higher generalizations of linear independence)?
Aside from algebraic independence, the other thing that comes to mind is the concept of "matroid", which is the sort of abstract general structure linear independent sets of vectors have.
Jun
13
comment Count number of colorings of tetrahedron, where colorings are indistinguishible if one can be reached from another by rotation
Try googling Burnside's Lemma or searching for answers on this site that apply it to counting problems. I've written a couple, but can't write out a full hint/toy example now.
Jun
13
comment The disc of convergence of a power series
"Geometrically" is alluding to a limit comparison test (or similar) with a geometric series which is known to converge.
Jun
11
comment Principle of mathematical induction
Roughly: Induction is an axiom if you use PA to define the natuals, and a theorem if you're using a set theoretical basis. The fact that there are different axiom systems that can make sense and lead to a certain conclusion you find obvious is a fact of life in mathematics.
Jun
10
comment Trajectories in orthogonal systems
The orbits are only circles in 2d when the eigenvalues have zero real part. Nonreal eigenvalues isn't enough.
Jun
5
comment Why can the transformation derived from a list of points and a list of their transformed counterparts not be affine or linear?
@null Absolutely right. As long as the three initial points don't lie on the same straight line, you will be able to solve the 6 equations for the 6 unknowns.
Jun
5
comment Why can the transformation derived from a list of points and a list of their transformed counterparts not be affine or linear?
It might be good to edit in more of the context so that your question does not depend on those links never breaking.
May
31
comment Find least numerator and denominator for a given sequence of numbers in decimal form
If there is a nice algorithm, it may be related to continued fractions. If you just want a handful of specific cases, things like wolframalpha.com/input/?i=Rationalize%5B0.1234444%2C.0000001%5D might work for you.
Mar
31
comment Can $[0,1]$ be partitioned with the following property?
I may be misreading something. If the measure of A is less than 1/2, isn't there no hope?
Mar
30
comment Infinite number of Derivatives
@user31415 My confusion was because my comment was about a different theorem. But as I said above, I now notice it's not needed for that other theorem either. I thank you for pointing that out.
Mar
30
comment Infinite number of Derivatives
@user31415 I didn't post an answer, nor did I use the word smooth. But nonetheless you're right. I think the proof I linked to doesn't make use of infinite differentiability in any way.
Mar
30
comment Infinite number of Derivatives
@user31415 I don't think you meant to @ me, but I agree.
Mar
29
comment Infinite number of Derivatives
And it's a bit less easy to prove that any function which is infinitely differentiable and has some derivative equal to zero at each point is still a polynomial. See the MathOverflow question
Mar
29
comment What's the difference between hyperreal and surreal numbers?
Things get a little hairy when dealing with class-sized hyperreal fields, but Keisler is quoted with a precise statement of "the surreals are the biggest version of the hyperreals" and proof sketch in Ehrlich's "The absolute arithmetic continuum and the unification of all numbers great and small"
Mar
29
comment Numerical system that includes the limit targets such as $0^+$, $0^-$, $1^+$ etc
You have listed the "limit targets", and I don't see why you want numerical operations on them, nor do I have much of an intuition about how they should be defined. Why must it be that $1^+=1+0^+$? As an aside (to complicate matters), a good way of thinking about limits is in terms of limsup and liminf from the left and right. Then at each real number $a$, a function defined on, say, $(a-h,a+h)$ has four "limits", which are extended reals.
Mar
28
comment proving that a function has a third derivative
The definition of $h(x)$ really doesn't say anything about $f$. The problem probably meant to say, at the very least, that $f$ was thrice differentiable, as otherwise the definition of $h$ may be invalid.